hpas 2017 MATHS p1

by

HPAS 2017 Mathematics Optional Paper-1: Complete Solutions

Welcome to the comprehensive solution guide for the Himachal Pradesh Administrative Service (HPAS) 2017 Mathematics Optional Paper-1. This resource provides detailed, step-by-step solutions designed specifically for civil service aspirants to master the core mathematical concepts and methodologies required for the exam.

Whether you are revising key theorems, practicing previous year questions, or mastering advanced analytical geometry and calculus, these carefully structured solutions will help streamline your preparation. Use the index below to jump directly to specific questions and topics.

Table of Contents

HPAS 2017 Maths Optional Paper-1 Question 1(a)

If A is a nilpotent matrix of index 2, show that A(I \pm nA) = A for any positive integer n.

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 1(b)

Examine the curve x=6t^2, y=4t^3-3t for concavity and convexity.

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 1(c)

Find the degree and order of the following differential equation:

\left|1+\left(\frac{dy}{dx}\right)^2\right|^{2/3} = \rho\frac{d^2y}{dx^2}

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 1(d)

If \vec{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}, then show that the vector \vec{r} is an irrotational vector.

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 2(a)

Find the matrix representation of a linear transformation t on V_3(\mathbb{R}) defined as t(x,y,z)=(2y+z, x-4y, 3x) corresponding to the basis B = \{(1,0,0), (0,1,0), (0,0,1)\}.

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 2(b)

State and prove the Cauchy-Schwarz inequality.

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 3(a)

Find the curve on which the three points of intersection of the curve x^2y - xy^2 + xy + y^2 + x - y = 0 with its asymptotes lie.

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 3(b)

Prove that a bounded function is not necessarily Riemann integrable.

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 4(a)

Find the condition that the straight line \frac{l}{r} = A\cos\theta + B\sin\theta may touch the circle r = 2a\cos\theta.

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 4(b)

Find the equation of a sphere which passes through the points (1, 0, 0), (0, 1, 0), and (0, 0, 1) and has the smallest possible radius.

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 5(a)

Solve the following differential equation by the method of variation of parameters:

(1-x)\frac{d^2y}{dx^2} + x\frac{dy}{dx} - y = (1-x)^2

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 5(b)

For Bessel’s function J_n(x), show that:
2nJ_n(x) = x[J_{n-1}(x) + J_{n+1}(x)]

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 6(a)

Obtain the Serret-Frenet formulas.

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 6(b)

Verify Stokes’ theorem for the function \vec{F} = z\mathbf{i} + x\mathbf{j} + y\mathbf{k}, where C is the unit circle in the xy-plane bounding the hemisphere z = \sqrt{1-x^2-y^2}.

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 7(a)

Five weightless rods of equal length are jointed together to form a rhombus ABCD with one diagonal BD. If a weight W is attached to C and the system is suspended from A, show that there is a thrust in BD equal to \frac{W}{\sqrt{3}}.

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 7(b)

Two forces act, one along the line y=0, z=0 and the other along the line x=0, z=c. As the forces vary, show that the surface generated by the central axis of their equivalent wrench is (x^2+y^2)z = cy^2.

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 8(a)

A particle moves in a curve such that its tangential and normal accelerations are equal and the angular velocity of the tangent is constant. Find the path.

Solution:

Enter solution here

HPAS 2017 Maths Optional Paper-1 Question 8(b)

A particle describes an ellipse under a force \frac{\mu}{(\text{distance})^2} towards a focus. If it was projected with velocity V from a point at a distance r from the center of force, show that its periodic time is

\frac{2\pi}{\sqrt{\mu}} \left[ \frac{2}{r} - \frac{V^2}{\mu} \right]^{-3/2}

Solution:

Enter solution here

Leave a Comment